Optimal control problem for speed at the end points in a vibration problem

In this study, processes relating to the control of speed at the end points for vibration problem have been presented. In the first section, after giving fundamental information about the wave equation, in the second section it has been referred from some definitions, theorem and lemma. In the next sections, proving the existence and uniqueness of the solution for vibration problem, some investigations have been carried out about the solution of the chosen optimal control problem. Moreover, the optimality conditions for the solution have been obtained.